Shape dynamics and Mach's principles: Gravity from conformal geometrodynamics

Abstract

In this PhD thesis, we develop a new approach to classical gravity starting from Mach's principles and the idea that the local shape of spatial configurations is fundamental. This new theory, "shape dynamics", is equivalent to general relativity but differs in an important respect: shape dynamics is a theory of dynamic conformal 3-geometry, not a theory of spacetime. Equivalence is achieved by trading foliation invariance for local conformal invariance (up to a global scale). After the trading, what is left is a gauge theory invariant under 3d diffeomorphisms and conformal transformations that preserve the volume of space. The local canonical constraints are linear and the constraint algebra closes with structure constants. Shape dynamics, thus, provides a novel new starting point for quantum gravity. The procedure for the trading of symmetries was inspired by a technique called "best matching". We explain best matching and its relation to Mach's principles. The key features of best matching are illustrated through finite dimensional toy models. A general picture is then established where relational theories are treated as gauge theories on configuration space. Shape dynamics is then constructed by applying best matching to conformal geometry. We then study shape dynamics in more detail by computing its Hamiltonian and Hamilton-Jacobi functional perturbatively. This thesis is intended as a pedagogical but complete introduction to shape dynamics and the Machian ideas that led to its discovery. The reader is encouraged to start with the introduction, which gives a conceptual outline and links to the relevant sections in the text for a more rigorous exposition. When full rigor is lacking, references to the literature are given. It is hoped that this thesis may provide a starting point for anyone interested in learning about shape dynamics.

117 pages, 2 tables, 10 figures, PhD thesis

References in corpus (28)

• Large N Field Theories, String Theory and Gravity
• Quantum Gravity at a Lifshitz Point
• The Dynamics of General Relativity
• The Holographic Weyl anomaly
• Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point
• On the Holographic Renormalization Group
• Forks in the Road, on the Way to Quantum Gravity
• The quantization of unimodular gravity and the cosmological constant problem
• Relativity without relativity
• The Link between General Relativity and Shape Dynamics
• The physical gravitational degrees of freedom
• AdS/CFT correspondence and Geometry
• The Superspace of Geometrodynamics
• Scale-Invariant Gravity: Geometrodynamics
• Constraints and gauge transformations: Dirac's theorem is not always valid
• Foundations of Relational Particle Dynamics
• Reconstructing AdS/CFT
• New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split
• Conformal Superspace: the configuration space of general relativity
• Interacting vector fields in Relativity without Relativity
• Triangleland. I. Classical dynamics with exchange of relative angular momentum
• Relational Particle Models. II. Use as toy models for quantum geometrodynamics
• The geometry of the Barbour-Bertotti theories II. The three body problem
• The geometry of the Barbour-Bertotti theories I. The reduction process
• Classical and Quantum Gravity on Conformal Superspace
• Triangleland. II. Quantum Mechanics of Pure Shape
• Implementing Mach's Principle Using Gauge Theory
• A First-Principles Implementation of Scale Invariance Using Best Matching

Cited by in corpus (12)

• Problem of Time in Quantum Gravity
• The Problem of Time and Quantum Cosmology in the Relational Particle Mechanics Arena
• Frequently asked questions about Shape Dynamics
• The role of time in relational quantum theories
• Classical Machian Resolution of the Spacetime Reconstruction Problem
• Poincaré invariance and asymptotic flatness in Shape Dynamics
• Machian Time Is To Be Abstracted From What Change?
• On Background Independence
• Kendall's Shape Statistics as a Classical Realization of Barbour-type Timeless Records Theory approach to Quantum Gravity
• Can Bohmian Mechanics Be Made Background Independent?
• Problem of Time: Facets and Machian Strategy
• Renormalizability, fundamentality and a final theory: The role of UV-completion in the search for quantum gravity